Let $A:D(A)\to H$ be a densely defined linear operator on a Hilbert space $H$, not necessarily continuous or self-adjoint. Let $v$ be a vector in $D(A)$.
If a solution to the differential equation $\dot\varphi(t)=A\varphi(t)$; $\varphi(0)=v$ exists, is it unique?
I am not asking for a result on the existence of solutions. It is easy to see that this question is equivalent to the case $v=0$, whereas the case of continuous $A$ follows from eg the Picard Lindelöf theorem.